We follow the same line of reasoning as in the proof of Lemma 5.7.18 in [1]}. Let \(x \in \overline{B(x^*,\rho )}\) . We use Lemma REF  for \(\delta = \eta /4\)  (and we let \(\rho \)  be small enough for Lemma REF  to hold). We construct a continuous path \(\psi ^x\)  with \(\psi ^x(0)=x\) , \(\psi ^x(t_x)=x^*\)  (\(t_x \le 1\) ) and \(I_{t_x,x}(\psi ^x)\le \eta /4\) . We then use Assumption REF  (D3). For \(T_1< \infty \) , we can construct a path \(\phi \in C[0,T_1]\)  such that \(\phi (0)=x^*\) , \(\phi (T_1)=z \in \widetilde{\partial O}\)  and \(I_{T_1,0}(\phi )\le \bar{V} + \eta /4\) . Subsequently, we use Lemma REF  and obtain a path \(\tilde{\psi }\)  with \(\tilde{\psi }(0)=z\) , \(\tilde{\psi }(s_x)\notin O\)  (\(s \le 1\) ), \(I_{s,z}(\tilde{\psi })\le \eta /4\)  and \(\operatorname{dist}(\bar{z},O)=:\Delta >0\) .Note that the Assumption (D5) is required here. We finally let \(\theta ^x\)  be the solution of the ODE (REF ) with \(\theta ^x(0)=\bar{z}\)  on \([0,2-t_x-s]\) , consequently \(I_{2-t_x-s,\bar{z}}(\theta ^x)=0\) , see Lemma REF .
